When Do Homomorphism Counts Help in Query Algorithms?

Authors Balder ten Cate , Victor Dalmau , Phokion G. Kolaitis , Wei-Lin Wu

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Balder ten Cate
  • University of Amsterdam, The Netherlands
Victor Dalmau
  • Universitat Pompeu Fabra, Barcelona, Spain
Phokion G. Kolaitis
  • University of California Santa Cruz, CA, USA
  • IBM Almaden Research Center, San Jose, CA, USA
Wei-Lin Wu
  • University of California Santa Cruz, CA, USA

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Balder ten Cate, Victor Dalmau, Phokion G. Kolaitis, and Wei-Lin Wu. When Do Homomorphism Counts Help in Query Algorithms?. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


A query algorithm based on homomorphism counts is a procedure for determining whether a given instance satisfies a property by counting homomorphisms between the given instance and finitely many predetermined instances. In a left query algorithm, we count homomorphisms from the predetermined instances to the given instance, while in a right query algorithm we count homomorphisms from the given instance to the predetermined instances. Homomorphisms are usually counted over the semiring ℕ of non-negative integers; it is also meaningful, however, to count homomorphisms over the Boolean semiring 𝔹, in which case the homomorphism count indicates whether or not a homomorphism exists. We first characterize the properties that admit a left query algorithm over 𝔹 by showing that these are precisely the properties that are both first-order definable and closed under homomorphic equivalence. After this, we turn attention to a comparison between left query algorithms over 𝔹 and left query algorithms over ℕ. In general, there are properties that admit a left query algorithm over ℕ but not over 𝔹. The main result of this paper asserts that if a property is closed under homomorphic equivalence, then that property admits a left query algorithm over 𝔹 if and only if it admits a left query algorithm over ℕ. In other words and rather surprisingly, homomorphism counts over ℕ do not help as regards properties that are closed under homomorphic equivalence. Finally, we characterize the properties that admit both a left query algorithm over 𝔹 and a right query algorithm over 𝔹.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and databases
  • query algorithms
  • homomorphism
  • homomorphism counts
  • conjunctive query
  • constraint satisfaction


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